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NASA TECHNICAL TRANSLATION NASA TT F-15,955 r ■■ ENERGY FLOW IN ELECTROMAGNETIC WAVE FIELDS H. Landstorfer, H. Liska, H. Meinke and B. Miiller (BASA-TT-P-15955) ENEEGY FLOW IN t: .'-TROMAGNETIC wave fields (Kanner (Leo) A.-..ociat6s) 20 p HC $4.00 CSCL 2CN N7U-34596 Unclas G3/07 50372 Translation of "Energlestromung in elektromagnetlschen Wellenfeldern," Nachrichtentechnische Zeltschrlft (NTZ-Communicatlons Journal), Vol. 25, No. 5, 1972, pp. 225-231 I \.--- NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D.C. 205't6 OCTOBER 197^ ■ xs. *«« &,i«^*s^s&*ye'34yKj*iK^^v'is *ianJW**Bi-i*W^f^'.f(*^8*»>j;**,^'VWT*»*^-»-">^ ENERGY FLOW IN ELECTROMAGNETIC WAVE FIELDS H. Landstorfer, H. Llska, H. Melnke and B. Muller Preliminary Remark /225 * In his field equations. Maxwell chose to represent field behavior by means of field strengths. This undoubtedly was due to historical reasons. We are of the view that field strengths provide more cf a formal mathematical description of the state of the field, whereas the distribution, motion and conversion of energy (conversion of electrical energy Into magnetic energy and vice versa) represent the actual physical process. It appears possible that better Information concerning the behavior of energy may also result in an Improved understanding of electromagnetic processes. A better physical understanding would assist the engineer In all of those complex cases In which mathematical com- putation Is not feasible or Is no longer reasonable because of the high mathematical outlay. Unfortunately, a great many prac- tical applications involving electromagnetic waves belong among the noncalculable cases, due to the severe boundary conditions and have only been amenable to experimental research so far. 1. Instantaneous Energy Flow The Poyntlng vector, called the vector of energy flow den- sity in the following [5a], Indicates the direction of energy flow, via its own direction, and energy transport per unit time, re- ferred to the unit of area, via its length. In time-variable fields, it is defined for the time being as the tlme-vai'lable vector s of instantaneous energy flow density * Nrmbers in the margin indicate pagination in the foreign text, e X h (1) As shown in Pig. 1, it is perpendicular to the vector e of in- stantaneous eirctric field strength and to the vector h of instantaneous magnetic field strength. Its length is s = Ih 'COS a (2) where a is the angle between the spatial vectors e and h. Our studies on instantaneous energy movement [4] have shown that very complex and difficultly understood energy behavior occurs even in relatively simple cases. Moreover, all time-variable field pro- cesses have the basic disadvantage that they can be represented only through a series of pictures, e.g. instantaneous patterns; the result actually becomes completely recognizable only as a motion picture. The presentation of such a sequence of pictures in a book or Journal is expensive and yet not very satisfactory, ihe reader is referred to our instantaneous patterns of the near field of an antenna in [1] and [2], which are considerably clearer in effect in a motion picture which has been made. 2. Time-Averaged Energy Flow Plotting the vector of time-averaged energy flow density, which thus indicates the mean of energy transport over time at a given location in terms cf direction and magnitude, has proven to be particularly simple and informative. This method was intro- duced by Wolter [3]. At each point in the field, there is thus a vector, constant over time, which indicates the direction of mean energy flow and its mean magnitude at this point. Actual energy movement is the result of superimposing an oscillation over this mean movement; cf. Section 3- We can then t-how stream- lines for time-averagtd energy flow in this space on the basis of known principles of flow theory; the vectors of time-averaged energy flow density are tangents to the streamlines. Such a V Fig. 1. Vector of energy flow density (Poyntlng vector). streamline pattern Is very similar to stationary flow In a source-free fluid. We obtain time-averaged energy transport by means of a generalized pattern of effective values with the aid of complex amplitudes. In a rectan- gular coordinate system (Fig. 2) , complex electric field strength vector E has the complex components lz = Ex-ei^'; £y = £^-eiv.; £, = £,• el»". ( 3) Complex magnetic field strength vector H has the complex components Fig. 2. Vectors of a TM wave. Hr = Hx- eif; Hy= Hy- eJ^«; H^^ H^- e'". . ( 4 ) \ As an expansion of equation (1), we can define a complex vector for energy flow density [6] S=SR + jSi = y(ExH*) (5) H* is the complex conjugate vector of magnetic field strength. /226 This definition of energy flow density is a generalization of the formula for complex apparent power, in which ono of the two co- efficients also appears with its complex conjugate value. The coefficient 1/2 always occurs in effective values when field strengths are given in peak values. The transport of real power is described by the real part, S|^. Sj^ is the actual energy flow density, which also establishes V. the direction of streamlines. The direction of Sr is the mean propagation direction of the energy. The imaginary part Sj from equation (5) represen;s a reactive power density and describes an oscillating movement of energy about its time-averaged movement (see Section 3) . The formula for real part Sr Is relatively extensive for general three-dimensional wave processes, so two simpler examples will merely be considered formally in the following. Fig. 2 shows the first case. Here, the electric fields of the wave lie in an x-z plane, and magnetic field Hy is perpendicular to the plane (TM wave). Fig. 3 shows the second case, in which magnetic field lies in a plane with components H^^ and H^;, and electric field is perpendicular to this plane as an Ey component (TE wave). From the convention in Fig. 1, with a = 90°, components E;c and Hy from equations (3) and {k) form the component Srz of time-averaged energy flow density in Fig. 2: -Sr, — -^ExUy- cos (v»z — tpv) (6) and components Eg and Hy form the component Sr^ of energy flow; SVLX = —-^EtHy- COS (y, - <py) (7) Pig. 3. Vectors of a TE wave. The (mathematically) positive directions of the components of field strength and energy flow den- sity are diagrammed in Fig. 2. If we apply the convention in Fig. 1 to component Sr^, then, according to equation (7), Srx must be shown in the negative direction. \> •s In Fig. 3, components Ey and H^^ form the component S^^ of energy flow: likewise, components Ey and Hg form the component Srx of energy flow: 1 (Q) In these two special cases, the energy streamlines lie in the plane given and thus can be represented graphically by simple means. With the exception of a three-dimensional case in Fig. 13> only planar cases will be represented in the following. As is conventional for planar flows, the streamlines are drawn at dis- tances such that equal energy flows move between each pair of adjacent streamlines. Thus the distance between streamlines in such pictures is a measure of flow density. A small interval indicates high flow density. 3. Ellipse of the Instantaneous Vector of Energy Flow It is shown in [7] that in the special cases shown in Figs. 2 and 3, instantaneous vector s(t) in equation (1) is obtained from the time-averaged value Sr in equations (6) through (9) with an additional time-dependent vector AsCt): 8(t) - Sj^ + A8(t). (10) This As follows an ellipse with time, as shown in Fig. i|, in such a manner that 8 becomes zero at a certain time. Sfj and Si are conjugate diameters of the ellipse, and the ellipse can be constructed from these two vectors by known methods. In a plane wave with linear propagation. •(0 = Se[1 +coa(«>< + i)]. (11) and the ellipse degenerates Into a straight line which has the direction of Sr. The ellipse thus describes the periodic oscillations superimposed on time-averaged energy flow. In a cer- tain way, the transverse oscillations represent a reactive power in the field. We suspect that this reactive power has an effect on the behaviors of streamlines for real power. Fig. 4. Ellipse of the complex vector of energy flow density. 4 . Energy Velocity If we wish to characterize the movement of energy in a wave field in detail, we must define an energy velocity. In a plane TEM wave, linear movement occurs everywhere and at all times and is at the velocity of light. In a more general wave field, energy /227 velocity , in terms of direction and magnitude, is a function of position and time. This applies to waves in homogeneous waveguides. The energy dW in an infinitesimal volume element dV will be called an energy packet. The energy packet moves at the in- stantaneous energy velocity associated with the particular location and time. A definition of velocity is not without problems, in that the identity of the energy packet as an individual cannot be determined along its path by measurement. A definition of velocity for unidentifiable individual entitles is always of a purely formal nature, but must be free of contradiction, i.e. must be compatible with all measured physical phenomena. In spite of thJs difficulty, we believe that the energy velocity defined by us can provide certain presentational advantages and makes certain verifiable conclusions possible, and thus is cer- tainly more than a pure formalism. A noncontradictory definition for a velocity vector is ob- tained if, in analogy to fluids, we define velocity as the quotient of energy flow density and energy density [4], If we use instantaneous values s for flow density and w for energy density here, we obtain the vector v of instantaneous energy velocity s/w. (12) If we use the time-averaged values Spj for flow density and Wjjj for energy density, we obtain mean energy velocity 'm Sr/W; m" (13) With the aid of instantaneous velocity v, we can calculate the path covered by an energy packet with time (instantaneous path; cf. Figs. 6, 7 and 9). With the aid of mean velocity, we obtain the mean path of the energy packet along the mean streamline, since vector Vm always has the direction of Sp. 5. Homogeneous and Inhomogeneous Waves between Conducting Planes Pig. 5 shows a plane TEM wave between conducting planes. The case is that shown in Pig. 2, with £2*0. Only Ex exists. The electric flux lines are parallel vertical lines in direction E. Since the magnetic flux lines are perpendicular to the plane of the drawing, as in Fig. 2, energy flow has only a z-component. The horizontal lines marked with arrows are the instantaneous and time-averaged streamlines. The streamlines have the same inter- val between one another, and mean flow density is the same across the entire waveguide cross section. 1 i Conductor : f ) I 1 ! i Conductor ^' Pig. 5. Energy stream- lines in a TEM wave. b) V -y -{ Path of an ^ l energy packet Y^ Streamline Pig. 6. TM wave between parallel conducting plane, a) Instanta- neous electric flux lines, b) Instantaneous and time-averaged paths of an energy packet, c) Mean energy flow and mean energy velocity. Fig. 6a shows the instantaneous electric flux lines of a TM wave be- tween parallel conducting planes [8]. This is a case of the type shown in Pig. 2 with time-dependent Oj^ and eg. Prom Fig. 1, with a » 90°, instan- taneous energy flow s is perpendicular to the instantaneous electric lines of flux. Since the lines of flux move, magnitude and direction of energy flow density are con- tinually changing. In- stantaneous velocity has components in the x- direction and z-direction. As an example. Fig, 6b shows a characteristic path of an energy packet which makes not only a longitudinal movement but also tranverse oscillations. Energy movement is sym- metrical with respect to the dashed center line, along which Sr ■ at all times and across which no energy therefore travels. Time-averaged energy flow has only a z-component, and the streamlines are The distribution of time- /228 straight lines in the z-direction. averaged energy flow over the cross section is not uniform, as the distance between adjacent energy streamlines shows in Fig. 6c. -1 A concentration of energy flow density occurs close to the bounding conducting planes. Tlme-aver£!,ed energy velocity Is a function of X. Fig. 6c shows the distribution of energy velocity Vj^g, defined by equation (13) > over the cross section on the right. Fig. 7 shows the Instantaneous magnetic flux lines of a TE wave between parallel conducting planes. This Is a case like that shown In Pig. 3» with time-variable hx and hg: e» = £„(x)-cos(a>t-/Jz); ^^^^ hx = Hg{x)'Cos(<ut-Pz); (15) hM'=H,(x)-sm{u)t-fiz). (16) Instantaneous energy flow s Is perpendicular to the In- stantaneous magnetic flux lines. It has the components «r = -eyh,'=:-^EyH^[i + co82(to«-/?z)]., (18) Sx varies at twice the freq ency but vanishes In the average over time. The component of Instantaneous energy flow s^ contains not only the alternating part cos 2(u)t - gz) but also a part Inde- pendent of time. Identical to the time average of energy flow S^^. Energy density w can likewise be resolved Into a DC part and an AC part, the DC part being Identical to mean energy w^: (19) I Instantaneous velocity has components vx In the x-dlrectlon (transverse oscillation) and v^ In the z-dlrectlon (longitudinal c) L 1 Streamline •^M/Path of an -- ^energy packet c) t movement). As an ex- ample, Pig. 7b shows th3 characterJ'^* •'" path of an erer-"/ paei- - with transvei-^e oscillation. b) Energy movement la sym- metrical wltii respect to the dashed center line, on which S^^^ ■ 0, I.e. only Srz exists. Time-averaged energy flow has only a z- component. The stream- lines are straight lines In the z-dlrectlon. Time averaged energy flow Is distributed non- unlformly over the cross section, as the distance between adjacent stream- lines In Fig. 7c shows. A concentration of energy flow density occurs at the center of the waveguide. Fig. 7c shows the distribution of mean energy velocity v„2 over the cross section on the right. The nonuniform distribution of energy velocity explains the familiar problems Involved in transmitting modulated signals with waveguides. Signals appear at the output of the waveguide with distorted modulation because they move faster at the inside, on the average, than at the outside. This effect is also referred to as the dispersion of a waveguide. Fig. 8 shows the ellipses of instantaneous flow density for a cross-sectional plane of the waveguide as an example for Pig. ^. Along the center line, the ellipse degenerates into a straight line, since Sji^^ " 0. Pig. 7. I'E wave between parallel con- ducting planes, a) Instantaneous magnetic flux lines, b) Instantaneous and time-averaged paths of an energy packet; c) Mean energy flow and mean energy velocity. 10 <3- — ^ Pig. 8. Ellipses of the complex vector of energy flow den- sity (Pig. H) In a TE wave between parallel conducting planes (Pig. 7). Fig. 9. Streamlines 3n a waveguide. Example of £;r. stantaneous energy path. Pig. 9 show the mean streamlines In an elbow and» In the form of a dashed line, an example of the Instantaneous path of an energy packet C^]. In the curved region, energy is "pushed" out- ward somewhat, on the average (by a sort of centrifugal force), and the in- stantaneous movement exhibits certain transverse oscillations there. Pig. 10a shows mean streamlines where a waveguide of parallel conducting planes widens abruptly, at a relatively low frequency, at which the higher modes are still quite aperiodic in inhomogeneity. Excitation is produced by a TEM wave arriving from the t. E is the direct f electric flux ] in the hcmogeneous regi-n. The energy flow behaves almost as in the potential flow of an incompressible fluid. cur-v^T' Fig. 101-) shows the mean streamlines at a higher frequency, at which a low-energy region develops at the lower wall. The upper corner apparently has a suction effect at higher frequencies, so energy flows upward there. The resultant low-energy region on the lower conductor draws the energy downward again, and thus transverse oscillations arise in the energy flow, not only trans- verse oscillations In Instantaneous energy flow, as in Fig. 9» but also transverse oscillations in time-averaged energy flow. 11 b) c) d) Pig. 10 Mean energy streamlines In a strip line which abruptly widens to width b. a) At a relatively low frequency (2b/> « 0.5), b) with a low-energy regie" C'b/Xo = 0.93), c) with an energy vortex (2b/X ^ O.96), d) with a multiple vortex (2b/X - 0-99). Fig. lOc shows an even higher frequency at which an energy vortex has developed [9]. Such vortices accompany very pronounced transverse oscillations in energy. These transverse oscillations can produce a series of sequential vortices. Fig. lOd shows energy streamlines in a multiple vortex. The vortices fill out each of the spaces which are freed of advancing energy by the transverse oscillations. A constant quantity of energy rotates in the vortex region, when averaged over time. 12 I In such complex wave fields, the Instantaneous movements are already becoming so difficult to follows [4] that they no longer provide any information. The mean streamlines, on the other hand, provide information on the behavior of inhomogeneity . For ex- ample, the vortex in Fig. 10c blocks the passage of energy in the region of the vortex, thereby reducing the effective cross section of the waveguide and lengthening the path of transported energy. The vortex can easily be detected with instruments through its effect on the inhomogeneity r'^f lection factor; see also [9]. 6. Near Field of Rod Antennas Pig. 11 shows energy flow for a short rod antenna which is standing on a conducting plane and is fed coaxially. The cy- llndrically syr.unetrical fields have the character shown in Fig. 2. The fields were calculated in [7] and [10]. While con- ventional theories assume unrealistic slot feed, realistic feed through a coaxial line of finite size is treated here for the first time in an exact theory. Only in this way does it become possible to show how energy let-.-es the coaxial line into space along the streamlines. Pig. 11a shows the distribution of time- average vector Sr of energy flow density, and Fig. lib shows the associated streamlines [12]. It can be see.i here that the rod antenna does not "radiate"; rather, the energy moving from bottom to top in the coaxial feed line is altered in its direction of movement by the energy distribution that builds up in the antenna's near field and is distributed within the surrounding space. This change in streauline direction results from trans- verse energy oscillations, visible only in the instantaneous movement of energy, but still not in the mean streamlines in the case of short rods. These instantaneous transverse oscillations, which represent a type of reactive power in the field, can be seen from the imaginary part Si of energy flow density from equation (5) and the ellipses of complex energy flow density. 13 f •-■' a) ^» ^* \ \ / ^ y Fig. 11. Mean energy flow of a short rod antenna of length VlO. a) Distribution of the vector Sr of time-averaged energy flow density; b) Energy streamlines associated with Fig. 11a; c) Ellipses of tiie complex energy flow density vector along three energy streamlines from Pig. lib. m As an example. Pig. lie shows these ellipses along three streamlines from Pig. lib. The dimensions of the ellipses perpendicular to the streamlines are a good measure of the magnitude of transverse oscilllations. The ellipses become flatter and the oscillations smaller with increasing distance from the antenna. If the rod is made longer, the transverse os- cillations become larger and also visible on the mean streamlines. Fig. 12 shows the streamlines in the region around a longer rod. Once the mean stream- lines oscillate visibly back and forth, as in Fig. 12, the far field pattern /230 moves away from the familiar vertical pattern of the short radiator. Pronounced transverse oscillations produce a lobing of the far field pattern. In the future it should be pos- sible to explain the behavior of a wave in the far field quantitatively. too, on the basis of transverse energy os- cillations in the antenna's near field. 7. Effective Area of Receiving Antennas Fig. 12. Mean energy streamlines of a rod antenna of length X. The near field of receiving antennas has been calculated in [11]. Representing the field with flux lines proves to be problematical and unclear here, since the flux lines require a time-variable three-dimensional mode of repre- sentation (superposition of a plane and a cylindrically symmetri- cal wave field). On the other hand, it is again simple and informative to describe the reception process with streamlines of time-averaged energy flow in this case [13]. Fig. 13 shows a short rod antenna above a conducting plane, acting as a receiving antenna. A complex resistance Z_ equal to the complex conjugate Impedance of the rod (internal resistance of source) exists between the base of the rod antenna and the con- ducting plane, acting as the dlssipator of received power. It is that dlssipator which draws the maximum possible (available) real power from the receiving antenna. A plane TEM wave passes along the conducting plane, from right to left in Fig. 13, with elec- tric field perpendicular to the conducting plane. Fig. 13a shows the distribution of the vector of time-averaged energy flow den- sity in a vertical plane through the radiator. From these vectors we obtain the streamlines shown in the vertical plane in Fig. 13b. Pig. 13b also shows the streamlines in the horizontal conducting plane in perspective. The linear intersection of the two planes shown is the direction of the arriving wave. There are boundary streamlines which divide the wave field into two regions. Two such boundaries are shown with dashed lines /231 15 8) -.■y-,: • •rs-''^' ~ ^ -^ • • /// / ^ ^ ' ' ' '/' 1 1 1 ' ' ' / ' M \ ' ' ' ," ^ \ \ \ ' 1 t f ^ ^ ^ " / / / / • X ^ ^ y y y y '^ " ^ - y'" — -" H luijii: Pig. 13. Mean energy flow of a short receiving antenna, a) Distribution of the vector Sr of time-averaged energy flow density in a vertical plane through the radiator. The boundary streamline is dashed in. b) Mean energy streamlines (the horizontal plane is shown in per- spective), c) Perspective repre- sentation of boundary streamlines. In the vertical plane in Fig. 13b, all those streamlines move into the base of the antenna which are located under the bo^undary. All streamlines above the boundary go past the antenna. The boundary thus separates that part of the wave which passes into the re- ceiving antenna. Like- wise Pig. 13b shows the dashed boundary in the horizontal plane. All of the boundary streamlines are shown together in perspective in Fig. 130. They ter- minate at point P, perpendicular to the line which passes through the base of the antenna and is in the direction of the arriving wave, and they delimit a certain space. All streamlines outside this space flow past the receiving antenna. The streamlines within the delimited area all terminate at the base 16 of the receiving antenna in dlssipator Z. The energy transported along these streamlines yields the received power. The boundary streamlines asymptotically approach straight lines parallel to the direction of the wave with increasing distance from the receiving antenna. The area delimited by them asymptotically approachs a cross section in the process which represents the effective area of the antenna , for the energy of the arriving plane wave passing through this area is absorbed by a receiving antenna with optimum dissipator Z, and this is the definition of effective area. We obtain Information regarding the shape of this effective area for the first time in this manner. The size of the effective area produced in this manner is equal to the values obtained theo- reticallyt to several percent, within the range of feasible numerical precision, e.g. 3X /16it for short rods. Effective area has been calculated for various rod lengths. The height of the effective area increases with increasing rod height and is greater than rod height. Its width decreases with increasing rod height [13]. The possibility of calculating the shape of the effective area is an interesting indication of the extensive new information about complex electromagnetic wave fields which can be obtained by calculating energy streamlines. The system of streamlines can be supplemented with a family of orthogonal curves, which we call "phase lines." These phase lines describe the time taken by the wave process along the streamlines. 17 REFERENCES 1. Llska, H., Meinke, H., and Mohr, C, "Wave separation from a conical antenna," Nachrlchtentech. Z. 23, 7^-79 (1970). 2. Landstorfer, F. , "Wave separation from rod antennas," Nachrlchtentech. Z. 23. 273-278 (1970). 3. Wolter, H., "The problem of the path of light In total reflection," Z. Naturforsch. 5A. 276-283 (1950). 4. Llska, H. , "Energy transport In electromagnetic wave fields," dissertation, Munich Engineering University, 1970. 5. Macke, W., Elektromagnetlsche Felder [Electromagnetic Fields], Section 13^, "The streamline pattern," 3rd edition, Leipzig, 1965. 5a. Ibid. , Section 6l4. 6. Meinke, H.H. and Gundlach, F.W., Taschenbuch der Hoch- frequenztechnlk [Handbook of Radio Frequency Engineering], Section H^l, 3rd edition, Berlin, Sprlnger-Verlag, 1968. 7. Landstorfer, F. , "Wave separation and energy transport in the near fields of rod antennas used for transmission," habilltatlon paper, Munich Engineering University, 1971. 8. Meinke, H., Einfuhrung in die Elektrotechnik hoherer Frequenzen [Introduction to Radio Frequency Electrical Engineering], Section II. 1, Vol. 2, 2nd edition, Berlin, Sprlnger-Verlag, 1966. 9. Llska, H. , " Experimental evidence of the two elementary types of energy vortices in wave fields," Nachrlchtentech. Z^ 23, 445-^'^8 (1970). 10. Landstorfer, F. , "Admittance and current distribution of linear cylindrical rod antennas," Arch, elektr. Ubertr. 23, 61-69 (1969). 11. Miiller, B., "Calculating the electromagnetic field about a receiving antenna," Arch. Elektron. t)bertr.-techn. 26 (1972) (In press). 12. Landstorfer, F., "Energy transjjort in the near field of rod antennas," Arch. Elektron. Ubertr. -techn. 26 (1972) (in press) . 13. Miiller, B., "Calculating energy flow ln..the near field of a receiving antenna," Arch. Elektron. Ubertr. -techn. 26 (1972) (In press) . 18